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| Mirrors > Home > MPE Home > Th. List > cnconst | Structured version Visualization version GIF version | ||
| Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.) |
| Ref | Expression |
|---|---|
| cnconst | ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconst2g 7223 | . . . 4 ⊢ (𝐵 ∈ 𝑌 → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵}))) | |
| 2 | 1 | adantl 481 | . . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵}))) |
| 3 | cnconst2 23291 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾)) | |
| 4 | 3 | 3expa 1119 | . . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾)) |
| 5 | eleq1 2829 | . . . 4 ⊢ (𝐹 = (𝑋 × {𝐵}) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))) | |
| 6 | 4, 5 | syl5ibrcom 247 | . . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹 = (𝑋 × {𝐵}) → 𝐹 ∈ (𝐽 Cn 𝐾))) |
| 7 | 2, 6 | sylbid 240 | . 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹:𝑋⟶{𝐵} → 𝐹 ∈ (𝐽 Cn 𝐾))) |
| 8 | 7 | impr 454 | 1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 TopOnctopon 22916 Cn ccn 23232 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-top 22900 df-topon 22917 df-cn 23235 df-cnp 23236 |
| This theorem is referenced by: xrge0mulc1cn 33940 cxpcncf2 45914 |
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